I. Moaddel Haghighi,1 N. Malossi,1, 2 R. Natali,1, 2 G. Di Giuseppe,1, 2 and D. Vitali1, 2, 3
1
School of Science and Technology, Physics Division, University of Camerino, I-62032 Camerino (MC), Italy 2INFN, Sezione di Perugia, Italy
3
CNR-INO, L.go Enrico Fermi 6, I-50125 Firenze, Italy (Dated: November 13, 2018)
An opto–electro–mechanical system formed by a nanomembrane capacitively coupled to an LC resonator and to an optical interferometer has been recently employed for the high–sensitive optical readout of radio frequency (RF) signals [T. Bagci, et al., Nature 507, 81 (2013)]. Here we propose and experimentally demonstrate how the bandwidth of such kind of transducer can be increased by controlling the interference between two–electromechanical interaction pathways of a two–mode mechanical system. With a proof–of–principle device operating at room temperature, we achieve a sensitivity of 300 nV/√Hz over a bandwidth of 15 kHz in the presence of radiofrequency noise, and an optimal shot-noise limited sensitivity of 10 nV/√Hz over a bandwidth of 5 kHz. We discuss strategies for improving the performance of the device, showing that, for the same given sensitivity, a mechanical multi–mode transducer can achieve a bandwidth significantly larger than that of a single-mode one.
PACS numbers: 42.50.Wk, 78.20.Jq, 85.60.Bt
I. INTRODUCTION
Optomechanical and electromechanical systems have recently shown an impressive development [1], and lately entered a quantum regime in which quantum states of nanogram–size mechanical resonators and/or electro-magnetic fields have been generated and manipulated [2– 10]. They have been also suggested and already em-ployed for testing fundamental theories [11, 12], and for quantum–limited sensing [13, 14]. Furthermore, nanome-chanical resonators can be simultaneously coupled to a large variety of different degrees of freedom and there-fore they can transduce signals at disparate frequencies with high efficiency [15–20], either in the classical and in the quantum domain. Reversible transduction between optical and radio–frequency/microwave (RF/MW) sig-nals is nowadays particularly relevant, both in classical and quantum communication systems, and first promis-ing demonstrations with classical signals have been re-cently achieved [21–23]. In particular the conversion of RF/MW signals into optical ones can be exploited for the high–sensitive detection of weak RF/MW signals, by tak-ing advantage of the fact that the homodyne detection of laser light can be quantum noise limited with near–unit quantum efficiency. This could avoid many of the noise sources present for low frequency signals and could be useful for example in radio astronomy, medical imaging, navigation, and classical and quantum communication. Bagci et al. [21] reported a first important demonstration of this idea with an optical interferometric detection of RF signals with 800 pV/√Hz sensitivity, and which could be improved down to 5 pV/√Hz in the limit of strong electromechanical coupling. In this device, weak RF sig-nals drive an LC resonator quasi–resonantly interacting with the nano–mechanical transducer, whose motion in-duces an optical phase shift which is then detected with quantum–limited sensitivity. A first application of an
opto-electro-mechanical transducer for nuclear magnetic resonance detection has been recently demonstrated [24]. In Ref. [21] the detection bandwidth depends upon the LC bandwidth and the electromechanical coupling [17, 21], and is an important figure of merit of such kind of transducers [25]. Finding systematic ways of in-creasing the detection bandwidth is of fundamental im-portance in many of the above–mentioned applications: for example, more radio-astronomical sources could be detected, while in communication networks RF signals could be faster detected and processed. Here we show with a proof-of-principle experiment that a viable way to increase the bandwidth of opto–electromechanical trans-ducers is to couple the LC circuit simultaneously to two (or more) mechanical modes with nearby frequen-cies, and suitably engineer the two electromechanical cou-plings in order to realise a constructive interference be-tween the two RF–to–optical signal transductions me-diated by each mechanical mode [see Fig. 1(a)]. Mul-timode optomechanical [26] and electromechanical [27– 29] systems have been recently studied and operated in a quantum regime, but here we exploit them with the aim of improving bandwidth and sensitivity of an opto-electro-mechanical transducer. The mechanical trans-ducer is a 1 mm × 1 mm SiN membrane of 50 nm thick-ness, coated with a 27 nm Nb film with a central circu-lar hole (Norcada Inc., see inset of Fig. 1), capacitively coupled through Cu electrodes to an LC resonator and operated at room temperature. The mechanical modes exploited are the split doublet (1, 2) − (2, 1) revealed by optical homodyne detection. The achieved sensitivity of the two–mode transducer is 300 nV/√Hz over a band-width of 15 kHz in the presence of RF noise, and the optimal shot–noise limited sensitivity is 10 nV/√Hz over a bandwidth of 5 kHz. The sensitivities are obtained in the case of electromechanical couplings for the two modes equal to G1 = 118.41 V m−1 and G2 = −115.31 V m−1.
Homodyne detector Local oscillator Spectrum analyser Vacuum chamber Membrane HWP PBS PBS HWP Lens Lens V VDC AC Si SiN Nb Cu
a)
b)
FIG. 1: a) Scheme of the RF–to–optical transducer and of the interference between the two transduction pathways through the two mechanical modes, x1, and x2. The two modes are simultaneously capacitively coupled with elec-tromechanical coupling G1and G2to the same LC resonator q, and eventually through a direct mechanical interaction, λ. At the same time the motion of the two resonators is readout by an optical interferometer using the light reflected from the membrane, δYout, with optical couplings α1 and α2. Since modulation of the phase noise of the optical beam occurs through two different paths (via mode 1 or mode 2), the sig-nal detected by the optical interferometer depends upon the interference between these two paths, which in turn can be controlled through the electrode configuration of the mem-brane capacitor. b) Experimental setup. An RF resonator is constituted by an inductor and a membrane capacitor placed in a vacuum chamber evacuated at 1 × 10−7mbar. The me-chanical displacement is revealed by homodyne detection of the light reflected by the membrane. The electromechanical coupling is controlled by applying a dc–bias VDC over two electrodes. The system is driven inductively, through two ca-pacitors, with an RF signal VAC using an antenna. Inset: a 1mm × 1mm SiN membrane coated with a 27 nm Nb film stands on top of four segment electrodes, forming a capacitor Cm(x) modulated by the membrane motion.
However, as we show in Sec. II, the method is general and could be exploited to reach larger bandwidths at a sensitivity comparable to that of single mode transduc-ers [21]. The paper is organized as follows. In Sec. II we introduce the multi-mode transducer theoretical frame-work. In Sec. III we show and discuss the experimental result showing the performance of our device, and we also
see how one can improve the design so that a two-mode transducer can achieve a larger bandwidth at the same sensitivity of a single-mode electromechanical transducer. Concluding remarks are provided in Sec. IV.
II. THEORETICAL FRAMEWORK
The system studied here is formed by a nanomechani-cal system capacitively coupled to an LC resonator. We generalize here to the multi-mode case the treatment of Ref. [21]. The nanomechanical system has a number of vibrational normal modes which can be described in terms of effective mechanical resonators with mass mi,
frequency ωi, displacement xi, momentum pi, so that
the effective Hamiltonian of the system is
H =X i p2 i 2mi +miω 2 ix2i 2 + φ2 2L+ q2 2C({xi}) − qV, (1)
where φ is the flux in the inductor, q is the charge on the capacitors, and V is the voltage bias across the capacitor. The coupling arises due to the displacement dependence of the capacitance C({xi}). This Hamiltonian directly
leads to the Langevin equations ˙ xi= pi mi , (2) ˙ pi= −miω2ixi− q2 2 ∂ ∂xi 1 C({xi}) − Γipi+ Fi, (3) ˙ q = φ L, (4) ˙ φ = − q C(x)− ΓLCφ + V, (5)
in which the terms corresponding to the damping rate Γi
of the i–th membrane mode, and to the resistive dissi-pation rate ΓLC = R/L of the LC resonant circuit have
been included, as well as driving forces Fiacting on each
membrane mode. Assuming that Fiare zero–mean
ther-mal Langevin forces, and writing the applied voltage as a large d.c. offset and a small fluctuating input
V (t) = VDC+ δV (t), (6)
we can linearize the Langevin equations around an equi-librium state of the system characterized by (¯xi, ¯pi, ¯q, ¯φ)
and satisfying the conditions
miωi2x¯i= − ¯ q2 2 ∂ ∂xi 1 C({xi}) x i=¯xi (7) = q¯ 2 2 ∂C({xi}) ∂xi x i=¯xi 1 C({¯xi})2 , ¯ q = VDCC({¯xi}), (8) ¯ pi= ¯φ = 0. (9)
The dynamical equations for the small fluctuations, pro-vided that the system is stable, are given by
δ ˙xi(t) = δpi(t) mi , (10) δ ˙pi(t) = −miωi2δxi(t) (11) −q¯ 2 2 ∂2 ∂2x i 1 C({xi}) x i=¯xi | {z } 2mωi∆ωi δxi(t) −q¯ 2 2 X j6=i ∂2 ∂xi∂xj 1 C({xi}) x i=¯xi | {z } λij δxj(t) − Γiδpi− ¯q ∂ ∂xi 1 C({xi}) x i=¯xi | {z } Gi δq(t) + Fi, δ ˙q(t) = δφ(t) L , (12) δ ˙φ(t) = − δq(t) C({¯xi}) − ΓLCδφ(t) + δV (t) −X j ¯ q ∂ ∂xj 1 C({xi}) x i=¯xi | {z } Gj δxj(t). (13)
Here, we have introduced the electro–mechanical cou-pling parameters Gi = ¯q ∂ ∂xj 1 C({xi}) xi=¯xi , (14)
the mechanical coupling between the vibrational normal modes induced by the second-order dependence of the capacitance upon the membrane deformation,
λij = ¯ q2 2 X j6=i ∂2 ∂xi∂xj 1 C({xi}) x i=¯xi , (15)
and the mechanical frequency shifts
∆ωi = ¯ q2 4miωi ∂2 ∂x2 i 1 C({xi}) x i=¯xi . (16)
Absorbing the frequency shifts into re-defined ωi and
transforming to the Fourier domain yields
−iΩ δxi(Ω) = δpi(Ω)/mi, (17) −iΩ δpi(Ω) = −miωi2δxi(Ω) − X j λijδxj(Ω) − Γiδpi(Ω) − Giδq(Ω) + Fi(Ω), (18) −iΩ δq(Ω) = δφ(Ω)/L, (19) −iΩ δφ(Ω) = −δq(Ω)/C − ΓLCδφ(Ω) + δV (Ω) − X j Gjδxj(Ω). (20)
These algebraic equations can be used to calculate the response of the system to excitations through a force or voltage drive. For notational convenience, we define the
susceptibilities χi(Ω) = 1 mi(ωi2− Ω2− iΩΓi) , (21) χLC(Ω) = 1 L (Ω2 LC− Ω2− iΩΓLC) . (22)
of the i–th mechanical mode and of the LC resonator, respectively, and we have defined the circuit resonance frequency ΩLC= (LC)
−1/2
. In the general case of many membrane modes the solution can be easily derived when λij = 0, i.e., without the direct mechanical coupling
me-diated by the capacitance and in the presence of only the indirect coupling through the LC resonator.
A. Two mechanical modes
We restrict now to the case of our system where the detection bandwidth includes only two mechanical modes and the effect of the other spectator modes is negligible (i.e., it falls below the noise level). Using Eqs. (17)–(22) one can write
χ1(Ω)−1δx1(Ω) = −λδx2(Ω) − G1δq(Ω) + F1(Ω), (23)
χ2(Ω)−1δx2(Ω) = −λδx1(Ω) − G2δq(Ω) + F2(Ω), (24)
χLC(Ω)−1δq(Ω) = −G1δx1(Ω) − G2δx2(Ω) + δV (Ω).
(25) From the last equation we have
δq(Ω) = −χLC(Ω) [G1δx1(Ω) + G2δx2(Ω) − δV (Ω)] ,
(26) and substituting in the first two we get
ξ1δx1(Ω) = β δx2(Ω) − G1χLCδV (Ω) + F1(Ω), (27) ξ2δx2(Ω) = β δx1(Ω) − G2χLCδV (Ω) + F2(Ω), (28) with ξi= χi(Ω)−1−G2i χLC(Ω) and β = [G1G2χLC(Ω)− λ] . Then we have (i = 1, 2) δxi(Ω) = ξ3−iFi(Ω) ξ1ξ2− β2 +β F3−i(Ω) ξ1ξ2− β2 − ξ3−1Gi+ βG3−i ξ1ξ2− β2 χLC(Ω)δV (Ω) , (29)
The signal detected by the optical interferometer is the phase quadrature δYout of the light reflected from the
membrane, which can be written in the frequency domain as
δYout(Ω) = α1δx1(Ω) + α2δx2(Ω) + δYin(Ω), (30)
that is, it is the sum of the vacuum phase noise, δYin(Ω),
and the displacement fluctuations of the two mechani-cal modes weighted by the optomechanimechani-cal couplings αi,
which depend upon the overlap of the selected membrane mode with the transverse profile of the optical field. We calibrate the output signal as a displacement spectrum, so that δY (Ω) has the same units of δxj(ω), that is,
m/√Hz. As a consequence the couplings αjcoincide with
the dimensionless transverse overlap parameters defined in Eq. (A6) (see Appendix A). Using the fact that the four noises, F1, F2, δV and δYinare uncorrelated, we can
write the output optical phase spectrum as the sum of four independent terms Sout(Ω) = α1ξ2+ α2β ξ1ξ2− β2 2 SF 1(Ω) + α2ξ1+ α1β ξ1ξ2− β2 2 SF 2(Ω) + Sin(Ω) + α1(ξ2G1+ βG2) + α2(ξ1G2+ βG1) ξ1ξ2− β2 2 |χLC(Ω)| 2 SδV(Ω), (31)
where SF j(Ω) = 2mjΓjkBT , j = 1, 2 are the
Brown-ian force noise spectra, with T the system temperature, SδV(Ω) is the noise voltage at the input of the LC circuit,
and Sin(Ω) is the optical shot noise spectrum. Let us now
try to readjust and rewrite this general expression for the detected spectrum in order to get some physical intuition from it. We first define the effective mechanical suscep-tibilities of the two modes, modified by the interaction with the LC circuit, (i = 1, 2)
χeffi (Ω) = ξ3−i ξ1ξ2− β2 , (32) χeff i (Ω) −1 = χ−1i (Ω) − G2iχLC(Ω) − β 2 χ−13−i(Ω) − G23−iχLC(Ω) . (33)
The detected spectrum of Eq. (31) can be then rewritten as Sout(Ω) = |α1+ α2µ2(Ω)|2 χeff1 (Ω) 2 SF 1(Ω) + |α2+ α1µ1(Ω)| 2 χeff2 (Ω) 2 SF 2(Ω) (34) + |I(Ω)|2|χLC(Ω)|2SδV(Ω) + Sin(Ω), where I(Ω) = α1χeff1 (Ω) [G1+ G2µ2(Ω)] + α2χeff2 (Ω) [G2+ G1µ1(Ω)] , (35) with µi(Ω) = β ξi = G1G2χLC(Ω) − λ χi(Ω)−1− G2iχLC(Ω) . (36)
It is evident from Eq. (34) that the transduction of volt-age input signals into the optical output signal is mainly determined by the quantity I(Ω) of Eq. (35), which is the sum of the two mechanical resonator contributions, i.e., the result of the interference between the two excita-tion pathways associated with each mechanical mode of Fig. 1(a). The quantity |I(Ω)| determines the voltage sensitivity of the transducer, and larger |I(Ω)| means higher sensitivity of our transducer, and therefore one has to engineer the couplings Gj in order to realise
con-structive interference between the transduction of the two modes and maximise |I(Ω)|.
The explicit expression of the detected spectrum at the output of the transducer simplifies considerably in the following limit: i) λ = 0 (which we have verified is actually satisfied by our experimental setup with a very
good approximation); ii) we stop at first order in Gi,
i.e., we neglect second order terms in Gi. In this limit
χeffi (Ω) → χi(Ω), µi(Ω) = 0 and one has the following
much simpler output spectrum, and a simpler form of the interference function I(Ω) in particular,
Sout(Ω) = |α1|2|χ1(Ω)|2SF 1(Ω) + |α2| 2 |χ2(Ω)| 2 SF 2(Ω) + Sin(Ω) (37) + |α1G1χ1(Ω) + α2G2χ2(Ω)| 2 |χLC(Ω)| 2 SδV(Ω).
The amplitudes and the relative signs of the couplings G1, α1, G2, and α2 determine the output spectrum and
therefore the behavior of the transducer itself. In fact, since the two effective mechanical susceptibilities χj(Ω)
halfway between the two mechanical resonance peaks are real and with opposite signs, we see from Eq. (37) that the products α1G1 and α2G2 must have the same sign
in the case of destructive interference, and opposite signs in the case of constructive interference between the two transduction pathways. In the first case we would ob-serve a spectrum region where the RF signal is canceled out by the destructive interference between the trans-duction of the two mechanical modes. In the second case we would observe a spectrum region between the two resonance peaks where the output signal is flat, and en-hanced by the constructive interference between the two electromechanical couplings. Experimentally we observe both behaviors and we refer to Sec. III for further details and discussion.
B. Relation between sensitivity and bandwidth
The transducer voltage sensitivity can be quantified by appropriately rescaling the detected noise spectrum of Eqs. (34) and (37), i.e., by defining the spectral voltage sensitivity as [21] q Sout δV (Ω) = pSout(Ω) |I(Ω)| |χLC(Ω)| . (38)
One expects that the better the sensitivity the narrower the corresponding bandwidth; we confirm that this is, in fact, the case, by quantifying this trade-off with a simple formula valid for the constructive interference case. Eq. (38) shows that the minimum detectable volt-age signal corresponds to the situation of minimum out-put noise Sout(Ω), and maximum value of the product
|I(Ω)| |χLC(Ω)|. Minimum output noise is achieved when
the contribution of all technical noises, that is, thermal and RF ones, are negligible with respect to the unavoid-able shot noise contribution, i.e., when the first, second and fourth term in Eq. (37) are negligible with respect to the third term, so that Sout(Ω) ' Sin(Ω). The
de-nominator of Eq. (38) is instead maximum when the LC circuit resonance peak (which is typically much broader than the mechanical peaks) is centered between the me-chanical doublet, and when |I(Ω)| is largest, showing why constructive interference is needed for a sensitive trans-ducer. Actually, |I(Ω)| is maximum exactly at the two mechanical resonance frequencies, where in principle one could get the best sensitivity. However, in order to get a physically meaningful and practical estimation of the optimal detectable voltage, we make here a conservative choice and consider the flat response region obtained by constructive interference between the two mechanical res-onances. In fact, the latter resonances do not represent a convenient working point because they are very narrow and extremely sensitive to small frequency shifts, and one expects a quite unstable transducer response there (see also the experimental results in the next Section). There-fore we take as optimal detectable signal the expression of Eq. (38) when Sout(Ω) ' Sin(Ω), evaluated halfway
be-tween the two mechanical resonances, at ¯Ω = (ω1+ω2)/2,
q SδVopt( ¯Ω) = p Sin( ¯Ω) |I( ¯Ω)|χLC( ¯Ω)| . (39)
In the simple case of a symmetric electromechanical sys-tem, that is, by assuming same masses (m = m1= m2),
electromechanical couplings (G = |G1| = |G2|), and
damping rates (Γ = Γ1 = Γ2) for the two
mechani-cal modes, and assuming also optimal optimechani-cal detection (α = α1= α2= 1), I( ¯Ω) is given by |I( ¯Ω)| = G m ¯Ω 1 iΓ + ∆Ω− 1 iΓ − ∆Ω , (40)
where ∆Ω = ω2−ω1. In typical situations one has ∆Ω
Γ, so that one can safely write |I( ¯Ω)| =
2G
m ¯Ω ∆Ω
, (41)
and replacing this latter expression into Eq. (39), one finally gets the desired sensitivity–bandwidth–ratio limit
q SoptδV ∆Ω = m ¯ΩpSin( ¯Ω) 2|G χLC( ¯Ω)| . (42)
This relation shows that, as expected, there is a trade-off between the voltage sensitivity and the bandwidth for an opto-electro-mechanical transducer with a given set of parameters. We can also see that, for a given shot noise level and fixing a desired voltage sensitivity SδVopt, one can increase the bandwidth either by decreasing the mechan-ical resonator mass, or by increasing the electromechani-cal coupling, always keeping the LC circuit at resonance so that |χLC| is maximum.
III. EXPERIMENT
A schematic description of the experiment is given in Fig. 1(b). A laser at 532 nm is split into a 10 mW beam (local oscillator), and a few hundreds µW one for prob-ing the mechanical oscillator. The beam reflected by the membrane is superposed to the local oscillator for de-tecting the phase–fluctuations. The low–frequency re-gion of the voltage spectral noise of the homodyne sig-nal is exploited to lock the interferometer to the grey fringe (i.e., in the condition where the interferometer output is proportional to the membrane displacement) by means of a PID control. The thermal displacement of the metalised membrane modes are revealed in the high–frequency range, as shown in Fig. 2. Calibration and fitting of the zoomed spectra of the fundamental and doublet modes shown in Fig. 2 allows us to ob-tain the optical masses m(1,1)opt , m(1,2)opt and m(2,1)opt of each mode. As explained in Appendix A, they are given by m(i,j)opt = meff/α(i,j)2, that is, by the effective mass of
these three membrane modes, which are all equal, divided by the square of the respective optomechanical coupling. One can estimate from them the most likely value of the center of the laser beam (and therefore of the optome-chanical couplings α(i,j)), and of the effective mass. The
latter is equal to meff ' 67.3 ng, in very good agreement
with the prediction of finite element method (FEM) nu-merical analysis of the metalised membrane.
The membrane is placed on top of a four–segment copper electrode to form a variable capacitor Cm({xi}),
which depends upon the transversal displacement of the membrane and therefore on the two mode displacements, xi. The distance h0 between the metalized membrane
and the four–segment electrodes has been determined to be 31.0(1)µm by the measurement of the frequency shift of the membrane fundamental mode (1, 1) as a function of the applied VDC, and the estimation of the effective area
of the membrane capacitor (see Appendix B and [21]). This capacitor is added in parallel to the rest of capac-itors of the LC circuit C0, and the total capacitance,
C({xi}) = C0+ Cm({xi}), is connected in parallel to
a ferrite core inductor with inductance of L ' 427µH. Taking into account the total series resistance of con-tacts and wires R, we have an LC resonator with res-onance frequency ΩLC/2π ' 383 kHz, therefore quasi–
resonant with the two mechanical modes, and a quality factor Q ' 81.
We have studied the behaviour of our device as a high– sensitive optical detector of RF–signals by fixing the ap-plied dc–bias at VDC = 270 V. A broadband RF signal
was injected into the system inductively using an auxil-iary inductor in front of the main LC inductor, and the corresponding displacement spectral noise (DSN) was de-tected. The nonzero voltage bias couples the LC circuit with the two mechanical modes whose motion, in turn, modulates the phase of the light; as a result, the input RF signal is transduced as an optical phase modulation
FIG. 2: Top: Calibrated displacement spectral noise (DSN) obtained from the homodyne measurement of the optical out-put when the system is driven by thermal noise only, i.e., without any electromechanical coupling. Top–left, DSN of the first mode (light–red dots) and the theoretical curve (red line) obtained with the best–fit values ω(1,1)m = 2π × 271.269 kHz, Γ(1,1) = 2π × 0.9 Hz, and m
opt(1,1) = 70.0(2) ng. Top–right, the mode doublet exploited for the transduction, with best–fit values ωm(1,2)= 2π × 382.69 kHz, Γ(1,2)= 2π × 4.9 Hz, m(1,2)opt = 1.73(1)µg, and ω(2,1)m = 2π×387.836 kHz, Γ(2,1)= 2π×2.6 Hz, m(2,1)opt = 1.18(1)µg (see Appendix A). Middle: Relative error between the detected frequencies (obtained from the peaks within the broader homodyne spectrum shown at the bot-tom of the figure) and those obtained from a numerical finite element analysis of the vibrational modes of the metalized membrane. The relative error found for the first five detected modes is less than 1%, and for the higher modes less than 3%. The black crosses indicate modes which are uncoupled to the light beam and therefore unobservable. Bottom: Broad homodyne spectrum. Each peak is associated with the cor-responding vibrational mode shape obtained from the finite element analysis. Light–grey and dark–grey curves denote shot and electronic noise contributions, respectively.
readout by the interferometer. The results are shown in Fig. 3, where the two plots correspond to two different electrode configurations.
The comparison evidences that by changing the elec-trodes on which the bias voltage is applied we are able to control the interference between the two transduc-tion pathways associated with each mechanical mode schematically illustrated in Fig. 1. In particular, by changing the electrodes, we are able to change the
ef-fective areas of the membrane capacitor, thereby chang-ing the relative sign between the two electromechanical couplings G1 and G2 (see Appendix B). As discussed
in Sec. II, this relative sign flip corresponds to switch-ing from constructive interference (Fig. 3a) to destruc-tive interference (Fig. 3b). In fact, in our case the two optomechanical couplings αi are positive numbers
(see below), and G1 and G2 have the same sign in the
case of destructive interference (Fig. 3b) and opposite signs in the case of constructive interference (Fig. 3a). This fact is confirmed by the theoretical curves which best overlap with the experimental data, correspond-ing to the followcorrespond-ing values of the electromechanical cou-plings: G1 = 118.41 V m−1 and G2 = −115.31 V m−1
for the red–line in Fig. 3a, and G1 = 117.63 V m−1 and
G2 = 110.89 V m−1 for the green–line in Fig. 3b. These
values have been confirmed within an 8% error, with an independent method based on the explicit evaluation of the membrane–electrode capacitance and its derivatives from the knowledge of the device geometry (see Appendix B and Ref. [21]). This geometrical estimation of the elec-tromechanical coupling crucially depends upon the over-lap between the electrodes and the positive and negative portions of the chosen membrane vibrational eigenmode, and therefore also provides an idea of how one can con-trol the relative sign between the two electromechanical couplings by applying the voltage bias to different elec-trodes.
The position of the laser beam with respect to the membrane determines the transverse overlap between the optical laser field and each mechanical mode, and there-fore the optomechanical couplings αi giving the weight
of the two interference pathways. As shown in Appendix A, we have found for the constructive interference case the best values α1 = 0.196 and α2 = 0.240, while for
the destructive interference we found α1 = 0.196 and
α2 = 0.121. The theoretical prediction is less accurate
away from the mechanical resonances for the destructive case; in this latter case in fact one has constructive inter-ference effects between the doublet modes and the fun-damental and higher mechanical modes, which are not fully taken into account by our model.
We remark that the possibility to tune the performance of our two–mode transducer by controlling the relative sign of the electromechanical couplings and the associ-ated interference effect is available only when the two mechanical modes are simultaneously coupled to two dis-tinct electromagnetic modes. In fact, if we would have simplified the scheme and used a unique electromagnetic (either radio–frequency or optical) mode both for cou-pling the modes and reading out the signals, we would always get a destructive interference pattern in the out-put spectrum, and the constructive case of Fig. 3a would be impossible. In such a case αi and Gi (i = 1, 2) share
the same sign, and therefore the response of the two me-chanical modes in the frequency band within the two res-onances would always be out of phase [30, 31]. Our two– mode transducer has analogies with the devices recently
a) b)
FIG. 3: Displacement spectral noise (DSN). Light–red and light–green dots correspond to the detection of constructive (a) and destructive (b) interference between the two mechani-cal transduction pathways, respectively, in the presence of an applied voltage bias VDC = 270 V and an input RF signal. Light–blue dots refer to the optical output spectrum due to thermal noise and without any RF input to the LC circuit. Solid red and green lines are the theoretical expectations with-out noise. Dashed red, green, and blue lines account also for the shot–noise contribution.
proposed in Refs. [32–34] for nonreciprocal conversion be-tween microwave and optical photons, and demonstrated in Refs. [33, 35], in which two mechanical modes are si-multaneously coupled via four appropriate drives with two different microwave cavity modes, for nonreciprocal signal conversion between the latter. In our case the con-figuration corresponding to the constructive interference of Fig. 3a realises the unidirectional transduction of RF signals into optical ones, while the one corresponding to the destructive interference of Fig. 3b realises an isolator which, within the bandwidth where I(Ω) ' 0, inhibits the transmission of RF signals to the optical output. The de-vice demonstrated here has the advantage that it does not require driving with four different tones and the validity of the rotating wave approximation. Moreover, the de-vice is easily reconfigurable because one can switch from one configuration to the other by simply switching elec-trodes [36].
As we have discussed in Sec. II, under the condition of constructive interference the mechanical modes are re-sponsible for an improved transduction of RF signals into the optical output within the frequency band between the two mechanical resonances. Therefore we expect that un-der the conditions of Fig. 3a the device acts as a trans-ducer with an increased bandwidth. This is confirmed by Fig. 4 where we show the voltage sensitivity (VS) de-fined in Eq. (38), i.e., the minimum detectable voltage, corresponding to the total noise spectrum of Fig. 3a di-vided by the interface response function. The red–light circles corresponds to the broader band voltage sensitiv-ity of our transducer which is equal to 300 nV/√Hz over a bandwidth of 15 kHz between the two modes, obtained in the case when RF noise dominates over thermal and shot noise. The blue dots and lines instead correspond to the optimal sensitivity of our device, around 10 nV/√Hz over a bandwidth of 5 kHz, achieved in the opposite limit
FIG. 4: Voltage sensitivity (VS) of the RF–to–optical trans-ducer. Light–red dots correspond to the inferred voltage sen-sitivity of our transducer from the blue data of Fig. 3a, that is the square–root of the DSN divided by the interface response function, which is equal to 300 nV/√Hz over a bandwidth of 15 kHz. The light–blue dots represent the optimal sensitivity achieved by our device in the case of negligible RF noise, equal to 10 nV/√Hz over a bandwidth of 5 kHz, dotted–black line. Dashed and solid lines represent the corresponding theoretical expectations as in Fig. 3. In this latter case the sensitivity– bandwidth ratio is in agreement with the optimal limit given by Eq. (42).
when the contribution of input RF noise is negligible with respect to thermal and shot noise. In this latter limit, in the flat region between the two resonance peaks thermal noise is also negligible, and the data (blue dots) exactly satisfy the optimal sensitivity–bandwidth ratio of Eq. (42). Instead, the data (red dots) in the pres-ence of a non–negligible noise contribution from the LC circuit, corresponds to a larger value of the sensitivity– bandwidth ratio compared to the optimal value. For ex-ample in Fig. 4, the red data correspond to a sensitivity– bandwidth ratio 10 times larger than the optimal one achieved by the blue data.
For comparison, in Fig. 5 we show the minimum de-tectable voltage in the destructive interference case of Fig. 3b. Even though in the case of large RF noise we have a comparable sensitivity of ∼ 300 nV/√Hz to that of the constructive interference case, the situation is completely different in the regime of negligible RF input noise (blue dots and theoretical curve). As expected, in this latter case, the sensitivity significantly worsens be-tween the two mechanical resonances and the minimum detectable voltage tends to diverge in correspondence to the destructive interference condition where the device acts as an isolator with respect to the RF input. It is ev-ident that in the presence of destructive interference the device cannot be operated as an RF–to–optical trans-ducer and that a sensitivity–bandwidth ratio cannot be even defined here.
We also remark that the present transducer can also be treated as a radiofrequency amplifier, transforming a voltage input signal into a voltage signal at the output of the optical detector, but at much higher signal to noise
370 375 380 385 390 395
Frequency [kHz]
10-9 10-8 10-7 10-6 10-5V
S
[V
/
pH
z]
FIG. 5: Voltage sensitivity (VS) of the RF–to–optical trans-ducer in the presence of destructive interference, from the data of Fig. 3b. Light–green dots correspond to the inferred voltage sensitivity of our transducer, that is the square–root of the DSN divided by the interface response function, which is equal to 300 nV/√Hz over a bandwidth approximately equal to 5 kHz. The light–blue dots represent the sensitiv-ity achieved in the case of negligible RF noise, which tends to diverge at the frequencies where one has destructive interfer-ence and the device is not sensitive to the input RF signal. Dashed and solid lines represent the corresponding theoretical expectations.
ratio, with a given gain and a given input impedance. At the working point described here, and correspond-ing to Fig. 4 and 5, we have measured for our device a gain of 30 db at the mechanical frequencies, and a gain of 10 db in the frequency range between them. Moreover we have characterized the input impedance obtaining a value Zin= (51.2 + 19.5i) kΩ.
A. Improving the two-mode transducer
performance
Using the theoretical description provided in Sec. II we now see how much one could improve the performance of our transducer in the constructive interference configura-tion. The voltage sensitivities achievable in a device sim-ilar to that experimentally demonstrated here, but with tunable electromechanical couplings |G1| = |G2| = G
and frequency separation ∆νm= [ω (2,1)
m − ωm(1,2)]/2π are
shown in Fig. 6. We show the transducer voltage sen-sitivity as a function of the electromagnetic coupling G at a fixed mechanical mode frequency separation ∆νm
in Fig. 6(a), and versus the mechanical mode splitting at a fixed G in Fig. 6(b) in the case of negligible RF noise. The voltage sensitivity is calculated from Eqs. (34) and (38) considering the following experimental param-eters: equal effective mass meff = 67.3 ng, equal
me-chanical damping rates Γ = 2π × 3.6 Hz, equal optome-chanical couplings α1 = α2 = 0.194, an LC circuit
res-onating halfway between the two mechanical resonances with a quality factor Q = 81.5, and a shot noise level Sin = 1.8 × 10−29m2/Hz. In Fig. 6(a) the two
reso-nance frequencies are fixed at ω(1,2)m /2π = 381 kHz and
ω(2,1)m /2π = 385.5 kHz. The blue-solid line denotes the
electromechanical coupling in our device, G = 118 Vm−1, while the dashed-line denotes the electromechanical cou-pling G = 5 kV m−1, which is used to calculate Fig. 6(b). Fig. 6(a) shows that, as expected, both sensitivity and bandwidth can be increased by increasing the electrome-chanical coupling and that one can achieve sensitivities comparable to those of Ref. [21] over a larger bandwidth in the strong coupling regime where the LC and the me-chanical modes hybridize, which occurs in our case when G > 10 kV m−1. A feasible way to achieve these val-ues of the coupling is to decrease the distance d be-tween the electrodes and the metalized SiN membrane since the coupling scales as the inverse square of d, and this strong coupling regime could be achieved with a dis-tance d ' 3µm. Fig. 6(b) instead shows that even in a regime away from the strong coupling regime, the trans-duction bandwidth can be increased simply by increasing the mechanical mode frequency splitting. With a cou-pling G = 5 kV m−1, about a factor of 30 larger than the one showed by our device, a sensitivity of the order of 1 nV/√Hz is reachable over a bandwidth that depends essentially only upon the mechanical mode splitting. In practice, by improving the device demonstrated here, for example by operating at a membrane capacitor distance of around d ' 3µm in order to reach G ' 10 kV m−1, and increasing the mechanical mode frequency splitting by using a rectangular membrane of 0.9 × 1.1 mm sides, one could achieve the same sensitivity of 800 pV/√Hz of Ref. [21] over a larger bandwidth of 40 kHz.
IV. CONCLUSIONS
We have theoretically shown and experimentally demonstrated that one can engineer a constructive in-terference between two or more mechanical modes cou-pled to the same resonant LC circuit in order to increase the transduction bandwidth of an RF–to–optical trans-ducer with a target voltage sensitivity equal to that of the single mechanical mode transducer. We have presented here a proof–of–principle experiment with a first gener-ation device proving the reliability of the proposed tech-nique and its physical insight. We have seen that an im-proved version of the same device could outperform exist-ing sexist-ingle–mode opto–electro–mechanical transducer in terms of sensitivity and especially in terms of bandwidth. The proposed multimode transducer based on construc-tive interference is advantageous and more flexible with respect to the one based on a single mechanical mode. In fact, in single–mode opto–electro–mechanical transduc-ers bandwidth and sensitivity are strongly related and determined only by the electromechanical coupling. In the case of capacitive coupling, it is extremely hard to achieve very large values of such a coupling because the bias voltage and the membrane capacitor area cannot be too large, and it is hard to reach membrane
capac-a)
b)
FIG. 6: Theoretical prediction for the voltage sensitivity (VS) of an RF–to–optical transducer based on a two-mode mechanical resonator in the case of negligible RF noise with a shot noise level Sin = 1.8 × 10−29m2/Hz. Top: VS as a function of the frequency and of the electromechanical cou-pling G (assumed to be equal in modulus for the two modes). The other parameters have been chosen to be very close to those of our experimental device. The two vertically brighter features represent the mechanical mode resonance frequen-cies at ωm(1,2) = 2π × 381 kHz and ω
(2,1)
m = 2π × 385.5 kHz, with same damping rate Γ(1,2)m = Γ(2,1)m = 2π × 3.6 Hz, same effective mass meff = 67.3 ng, and same optomechan-ical coupling α1 = α2 = 0.194. The LC circuit resonates at ωLC= [ω
(1,2)
m + ω
(2,1)
m ]/2 with a quality factor Q = 81.5. The blue solid line denotes the electromechanical coupling in our device, G = 118 V m−1, while the black dashed line denotes the value G = 5 kV m−1 needed to obtain a mean voltage sensitivity of the order of 1 nV/√Hz over a bandwidth of 15 kHz. For larger G both the sensitivity and the bandwidth increase. Bottom: VS as a function of the frequency and of the mechanical mode separation ∆νm = [ωm(2,1)− ωm(1,2)]/2π evaluated for the same parameters as the plot above, and with a value of the electromechanical coupling G indicated by the black dashed line of the top figure. The vertical black dashed lines represent the two resonance frequencies chosen above. We see that one can achieve and maintain a voltage sensitiv-ity of around 1 nV/√Hz over a bandwidth which increases for increasing frequency separation between the two mechanical modes.
itor distances well below one micron. On the contrary, in multimode opto–electro–mechanical transducers in the
constructive interference configuration, for a given volt-age sensitivity, the bandwidth is mainly determined by the mechanical frequency splitting and therefore it can be significantly increased even without entering the strong electromechanical coupling regime.
Acknowledgments
We acknowledge the support of the European Commis-sion through the ITN–Marie Curie project cQOM, the FP7 FET-Open Project n. 323924 iQUOEMS, and the H2020–FETPROACT–2016 project n. 732894 “HOT”. We also thanks Norcada for providing us the Nb metal-ized SiN membranes.
Appendix A: Data analysis
1. Determination of mechanical parameters from thermal noise spectra
The mechanical properties of the membrane vibra-tional modes, that is, their resonance frequency, damping rate and mass, can be extracted from the measured ho-modyne spectra in the presence of thermal noise only, that is, in the absence of the electromechanical coupling, occurring when VDC= 0 and the RF signal is turned off.
For a generic harmonic oscillator of mass m, frequency ωm and damping Γ in the presence of thermal noise at
temperature T , the variance of its mechanical displace-ment, hx2i = k
BT /m ωm2, is related to the displacement
spectral noise (DSN) Sxx(ω) by the relation
hx2i = Z +∞ −∞ Sxx(ω) dω 2π = Z +∞ 0 ¯ Sxx(ν)dν , (A1) where Sxx(ω) = 2 m Γ kBT |m(ω2 m− ω2− iωΓ)|2 , (A2)
and defining ω = 2πν, ωm = 2πνm, and Γ = 2πγ, one
has ¯ Sxx(ν) = 1 πm 2 γ kBT |ν2 m− ν2− iνγ|2 . (A3)
The measured DSN, ¯S(m)xx (ν), is obtained from the
cali-bration of the voltage spectral noise SV V(ν) effectively
detected at the output of our optical interferometer, ¯
S(m)xx (ν) = SV V(ν) G2xV , (A4)
with the calibration factor GxV = λ/(2πVpp), where
λ = 532 nm is the laser wavelength used, and Vpp is the
peak–to–peak voltage value of the interferometer interfer-ence fringes. Then the measured DSN is fitted with the theoretical ¯Sxx(ν) of Eq. [A3] obtaining best–fit values
for ωm and Γ. Due to the effect of the optical
trans-duction (see Eq. (30)), for each mechanical mode the fit provides for the mass the value of what can be called
the optical mass m(n,m)opt , which is related to the physi-cal effective mass of each mode and the optomechaniphysi-cal coupling α(n,m) by the relation m
(n,m) opt = m (n,m) eff /α 2 (n,m).
The variance of the mechanical displacement hx2i is in-stead equal to the size of the step in the measured dis-placement noise (DN), that is, the marginal of the DSN, (see the blue curves in Figs. 7–8). We have performed such a fit for the fundamental vibrational mode of the membrane (1, 1) (see Fig. 7), and for the first excited vi-brational doublet (1, 2) and (2, 1) exploited here for our transducer (see Fig. 8).
271.15 271.25 271.35
Frequency [kHz]
10-30 10-28 10-26 10-24 10-22D
SN
[m
2/
H
z]
0 5 10 15 20 25D
N
[p
m
2]
FIG. 7: Displacement spectral noise (DSN) for the fundamen-tal mode (1, 1). The calibration parameter is Vpp = 2.7 V, and the best–fit values are ωm(1,1)= 2π × 271.269 kHz, Γ(1,1)= 2π×0.9 Hz, and m(1,1)opt = 70.0(2) ng. The size of the step in the displacement noise (DN) (blue curve), that is the marginal of the DSN, determines the variance of the mechanical displace-ment hx2i(1,1)
to be 24.18 pm2.
For the (1, 1) mode we obtained the best–fit values ωm(1,1) = 2π × 271.269 kHz, Γ(1,1) = 2π × 0.9 Hz, and
m(1,1)opt = 70.0(2) ng. The size of the step in the
displace-ment noise (DN) yielded hx2i(1,1) ' 24.18 pm2. For the
(1, 2) − (2, 1) doublet instead we obtained the best–fit values ω(1,2)m = 2π × 382.69 kHz, Γ(1,2) = 2π × 4.9 Hz,
m(1,2)opt = 1.73(1)µg, and ωm(2,1) = 2π × 387.836 kHz,
Γ(2,1) = 2π × 2.6 Hz, m(2,1)
opt = 1.18(1)µg. The variances
of the mechanical displacement are hx2i(1,2)' 0.397 pm2,
and hx2i(2,1)' 0.590 pm2.
2. Determination of the effective mass and of the optomechanical couplings
The effective mass meff associated with a vibrational
mode depends in general upon the mode volume, and in the case of a thin membrane it can be written as
m(n,m)eff = Z Z dxdy σ(x, y) u(n,m)(x, y)2, (A5) 382.55 382.75
Frequency [kHz]
10-30 10-28 10-26 10-24D
SN
[m
2/
H
z]
387.75 387.95Frequency [kHz]
0.0 0.1 0.2 0.3 0.4 0.5 0.6D
N
[p
m
2]
FIG. 8: Displacement spectral noise (DSN) for the doublet (1, 2) − (2, 1). The calibration parameter is Vpp= 2.8 V. The best–fit values are ωm(1,2) = 2π × 382.69 kHz, Γ(1,2) = 2π × 4.9 Hz, m(1,2)opt = 1.73(1)µg, and ω
(2,1)
m = 2π × 387.836 kHz, Γ(2,1) = 2π × 2.6 Hz, m(2,1)opt = 1.18(1)µg. The variances of the mechanical displacement are hx2i(1,2) ' 0.397 pm2
, and hx2i(2,1)' 0.590 pm2
.
where σ(x, y) is the average mass surface density of the membrane and u(n,m)(x, y) is the dimensionless
eigen-function of the vibrational mode with indices (n, m) [40]. As discussed in the previous subsection, the masses obtained from the fitted thermal noise spectra instead depend also upon the optomechanical couplings α(n,m)
which differ from one mode to another because the laser beam illuminates a certain spot on the membrane, where different modes have different displacement amplitudes. After calibration of the DSN, the couplings α(n,m)
coin-cide with the dimensionless transverse overlap parame-ters [40], given by α(n,m)(x, y) = Z L 0 dx0 Z L 0 dy0u(n,m)(x0, y0)I(x, y, x0, y0), (A6) where I(x, y, x0, y0) is the normalised intensity profile of a laser beam centered at (x, y), and L is the length of the side of the square membrane. In the case of our experiment, the theoretical value of Eq. (A6) can be analytically evaluated because we used a TEM00
Gaus-sian beam with waist w at the membrane position, and we have verified with a finite element method analysis that for the first three vibrational modes studied here the homogeneous membrane eigenmodes, u(n,m)(x, y) =
sin(nπx/L) sin(mπy/L), provide a very good approxima-tion. Assuming optical losses from clipping negligible, the domain of integration can be extended to the entire plane, and one gets from Eq. (A6)
α(th)nm(x, y) = e−w2(k2n+k 2
m)/8 sin(knx) sin(kmy) , (A7) where kn = n π/L, and km= m π/L, which depend upon
the unknown beam center (x, y).
One can get a very good estimate of the beam center position (x, y) (and therefore of the transverse overlaps
and of the physical effective masses m(n,m)eff ) in our setup, by applying a treatment analogous to that of Ref. [26]. For each of the three detected vibrational modes, the variance of the mechanical displacement hx2i(n,m)
pro-vides an indirect estimate (¯x, ¯y) of (x, y), because hx2i(n,m)= kBT m(n,m)opt ω2 nm = α2nm(¯x, ¯y) kBT m(n,m)eff ω2 nm , (A8)
from which one derives the experimental estimate α(ex)nm (¯x, ¯y) '
q
hx2i(n,m)m(n,m)
eff ω2nm/kBT , (A9)
which depends upon the measured quantities hx2i(n,m),
ω2
nm, and T , and the unknown effective mass of
the mode m(n,m)eff . However, since unm(x, y) =
sin(nπx/L) sin(mπy/L) is a very good approximation, Eq. (A5) yields m(n,m)eff = mT/4 independent of (n, m),
where mT is the total mass of the membrane. Moreover
we expect that for the fundamental mode α2
11(¯x, ¯y) ' 1
because the measured waist w = 53.2(4)µm is much smaller than L = 1 mm and the beam is centered very close to the membrane center. As a consequence, we can safely assume m(n,m)eff ' m(1,1)opt = 70.0(2) ng for the three modes in Eq. (A9), which is also consistent with the value obtained from Eq. (A5) and membrane specifica-tions (1 mm × 1 mm square, 50 nm–thick SiN membrane, coated with a 27 nm Nb film with a 300µm–diameter central circular hole).
We then construct the χ2quantity
χ2(x, y) =X n,m α(ex) nm (¯x, ¯y) − α(th)nm(x, y) 2 , (A10)
and minimize it over (x, y). The minimizing points (x0, y0) are the most likely points, and the
correspond-ing likelihood density function of where the beam is po-sitioned is given by [26] L(x, y) = 1 2πσ2 Y n,m e−
α(ex)nm (¯x,¯y)−α(th)nm (x,y) 2
2σ2 (A11)
with σ2= χ2(x
0, y0), and whose contour plot is shown in
Fig. 9. The corresponding best estimation of the trans-verse overlap for the modes are
α11= 0.980 α12= 0.196 α21= 0.240 , (A12)
yielding the best estimate for the physical effective mass of the three modes, m(1,1)eff ' m(1,2)eff ' m(2,1)eff ' 67.3 ng, within a 3% error, and confirmed by finite element method (FEM) numerical analysis.
Appendix B: The electromechanical couplings
As shown in Eq. (14) the electromechanical couplings Gi depend upon the explicit expression of the
capac-itance of the LC circuit and its dependence upon the transverse displacement associated with each vibrational normal mode of the membrane. We can write for the total capacitance C = C0+ Cm({xi}), where C0 is the
0.0
0.2
0.4
0.6
0.8
1.0
x/L
0.0
0.2
0.4
0.6
0.8
1.0
y/
L
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
1e 10
FIG. 9: Position estimates from the χ2minimisation, showing the most likely points.
capacitance of the LC circuit (including fixed and tun-able capacitors) acting in parallel with the membrane capacitance Cm({xi}). We have verified that in our case
C0 Cm({xi}), so that from Eq. (14) one can also write
Gi' − Vdc C0 ∂Cm({xi}) ∂xi x i=¯xi . (B1)
Following [41] and exploiting the geometry of our membrane–electrode arrangement, one can derive a the-oretical model of the capacitance Cm({xi}) based on a
quasi–electrostatic calculation, which allows to derive both the electromechanical couplings Gi and the
fre-quency shifts of Eq. (16), and satisfactorily reproduces the data.
As shown in Fig. 2 of the main text, the membrane ca-pacitor is formed by a four–segment electrode in front of the partially metallized membrane. Since the membrane– electrode separation h0 is significantly smaller than the
inter–electrode gaps, we can neglect the direct capac-itance between electrode segments; the capaccapac-itance is then given by the series of two local contributions, one associated with the positive electrode segments and the membrane in front of it, C+, and the second one
corre-sponding to the negative electrode segments, C−, i.e.,
Cm= 1 C+ + 1 C− −1 . (B2)
For the calculation of C± we assume that the curvature
of the membrane is sufficiently small so that we can take it to be locally flat. We also neglect edge effects, so that for symmetry, and assuming perfect alignment, we may model the membrane–electrode capacitance locally as that of conducting parallel plates. This local capaci-tance per area only depends upon the local membrane– electrode separation along the direction normal to the
plane defined by the electrodes, and we can write C±= Z Z dxdy ε0ξ±(x, y) h0+ δz(x, y) , (B3)
where the integral is taken over the membrane surface, ξ±(x, y) is a mask function that equals 1 for points in
the membrane plane that are metalized and overlap with the fixed positive or negative electrode, and is zero other-wise, δz(x, y) is the membrane displacement field relative to the steady–state configuration, and ε0 is the vacuum
dielectric constant. We can always expand this field in terms of the vibrational eigenmodes ui(x, y) introduced
in Eq. (A5)
δz(x, y) =X
i
βiui(x, y), (B4)
where being the eigenmodes ui(x, y) dimensionless, the
coefficients βi are canonical drum mode position
coordi-nates. With this notation, the derivatives appearing in the expression for the couplings of Eq. (B1), become
∂Cm({xi}) ∂xi x i=¯xi → ∂Cm ∂βi eq , (B5)
where “eq” means that the derivative should be evaluated at the static displacement equilibrium configuration of the membrane, δz(x, y) = 0. We have explicitly
∂Cm ∂βi eq = 1 C2 + ∂C + ∂βi +C12 − ∂C − ∂βi 1 C++ 1 C− 2 eq , (B6)
so that, using Eqs. (B3)–(B4), and inserting the results into Eq. (B1), one finally gets
Gi=
VDC0
C0h20
Aeffi , (B7)
where we have defined the effective mode area
Aeffi = O(1)+,i [O(0)+ ]2 + O−,i(1) [O−(0)]2 1 O(0)+ + 1 O(0)− 2 , (B8)
in terms of the quantities O±,i(j) ≡
Z Z
dxdyξ±(x, y)[ui(x, y)]j j = 0, 1. (B9)
The explicit values of the two electromechanical cou-plings G1 and G2 associated with the two mechanical
modes used for our transducer can be obtained from the knowledge of C0, VDC, the distance h0 and the various
integrals O(j)±,i. We have evaluated the latter integrals numerically from the calibrated image of the electrode and from the properly normalized finite element numeri-cal solution of the two vibrational eigenmodes, while C0,
and VDC are easily measured. The membrane–electrode
equilibrium distance h0instead has been evaluated from
the measurement of the mechanical frequency shift of the fundamental vibrational mode.
1. Derivation of the membrane–electrode distance
Eq. (16) shows that each mechanical mode is shifted quadratically as a function of the applied DC voltage. A measurement of this quadratic phase shift provides a quite accurate indirect method for the determination of the distance h0between the metalized membrane and the
electrode. In our case we have measured the frequency shift of the fundamental mode (1, 1) (see Fig. 10). De-noting with i = 0 the fundamental mode (1,1), recalling that C = C0+ Cm({xi}) with C0 Cm({xi}) so that
¯
q ' C0VDC, and using Eq. (B5) and that ω0= 2πν0, one
can rewrite Eq. (16) as
∆ν0= − V2 DC 16π2m effν0 ∂ ∂β0 ∂Cm ∂β0 eq , (B10)
where Eq. (B6) has to be used for the evaluation of ∂Cm/∂β0. It is possible to verify that
∂ ∂β0 ∂Cm ∂β0 eq '2ε0A eff 0 h3 0 , (B11)
where Aeff0 is the effective area for the fundamental mode, and one can write
ν(VDC) = ν0 1 − ε0A eff 0 8π2m effν02h30 VDC2 . (B12)
On the other hand we can fit the experimental data of Fig. 10 with ν(VDC) = ν0 1 − Λ 8π2m effν02 Vdc2 ; (B13)
where Λ is a fitting parameter. Using the best–fit values derived above, ν0 = 2.712 69 kHz and meff =
67.3 ng, the best–fit value of the parameter Λ = 105.2(9)µF m−2, and using Aeff0 = 0.3546 mm2, and 0=
8.854 × 10−12s4/m3/kg, the distance between membrane
and electrode is evaluated to be
h0= ε0Aeff0 Λ 13 ' 31.0(1)µm . (B14) 0 50 100 150 200 250 300
V
DC[V]
264 265 266 267 268 269 270 271 272ν
[k
H
z]
FIG. 10: Mechanical resonance frequency shift of the funda-mental mode as a function of the applied dc voltage VDC.
With this derivation of the membrane–electrode dis-tance h0, we can finally estimate the electromechanical
couplings G1 and G2 using Eq. (B7) once that the
effec-tive areas Aeff
i have been estimated using Eqs. (B8)–(B9).
For the mode–electrode configuration of Fig. (11), our numerical estimation gives the values of −0.0178 mm2 for Fig. (11a), 0.0189 mm2 for Fig. (11b), 0.0185 mm2 for Fig. (11c), and 0.0190 mm2 for Fig. (11d). These
values of the effective area can be understood from the fact that the blue and yellow lobes denote respectively the negative and positive parts of the vibrational mode function. In each of the four configurations one of the two electrodes has approximately the same overlap with the positive and negative lobes, yielding therefore a neg-ligible contribution to the effective area of Eq. (B8). The other electrode yields the main contribution to the effec-tive area which is therefore negaeffec-tive for Fig. (11a) and
positive for the other three cases, so that the upper con-figurations corresponds to the constructive interference case and the lower ones to the destructive interference case.
Using these values for the effective areas, and insert-ing C0 = 404 pF, Vdc = 270 V and h0 = 31.0µm
into Eq. (B7), we get G1 = 116.4 V m−1 and G2 =
−109.6 V m−1for the upper electrode configurations
cor-responding to the constructive interference case (see Fig. 11a and 11b). Instead we get G1 = 117.0 V m−1
and G2= 113.9 V m−1 for the lower electrode
configura-tions corresponding to the destructive interference case (see Fig. 11c and Fig. 11d). These values are in very good agreement with the values given in the main text and obtained as best-fit parameters of the measured out-put spectra.
[1] M. Aspelmeyer, et al., Rev. Mod. Phys. 86, 1391 (2014). [2] A. D. O’Connell, et al., Nature 464, 697 (2010). [3] J. D. Teufel, et al., Nature 475, 359 (2011). [4] J. Chan, et al., Nature 478, 89 (2011). [5] E. Verhagen, et al., Nature 482, 63 (2012). [6] A. H. Safavi-Naeini, et al., Nature 500, 185 (2013). [7] T. A. Palomaki, et al., Science 342, 710 (2013).
[8] R. Peterson, et al., Phys. Rev. Lett. 116, 063601 (2016). [9] J. B. Clark, et al., Nat. Phys. 12, 683 (2016).
[10] R. Riedinger, et al., Nature 530, 313 (2016). [11] I. Pikovski, et al., Nat. Phys. 8, 393 (2012). [12] M. Bawaj, et al., Nat. Commun. 6, 7503 (2015). [13] J. Chaste, et al., Nature Nanotech. 7, 301 (2012). [14] C. B. Møller, et al., Nature 547, 191 (2017).
[15] K. Stannigel, et al., Phys. Rev. Lett. 105, 220501 (2010). [16] C. A. Regal, et al., J. Phys. Conf. Ser. 264, 012025
(2011).
[17] J. M. Taylor, et al., Phys. Rev. Lett. 107, 273601 (2011). [18] Y.-D. Wang, et al., Phys. Rev. Lett. 108, 153603 (2012). [19] L. Tian, Phys. Rev. Lett. 108, 153604 (2012).
[20] S. Barzanjeh, et al., Phys. Rev. Lett. 109, 130503 (2012). [21] T. Bagci, et al., Nature 507, 81 (2013).
[22] R. W. Andrews, et al., Nat. Phys. 10, 321 (2014). [23] J. M. Fink, et al., Nat. Commun. 7, 12396 (2016). [24] K. Takeda, et al., arXiv:1706.00532 [quant-ph] (2017). [25] E. Zeuthen, et al., arXiv:1610.01099 [quant-ph] (2016). [26] W. H. P. Nielsen, et al., Proc. Natl. Acad. Sci. (USA)
144, 62 (2017).
[27] F. Massel, et al., Nat. Commun. 3, 987 (2012). [28] A. Noguchi, et al., New J. Phys. 18, 103036 (2016) [29] X. Han, et al., Phys. Rev. Lett. 117, 123603 (2016). [30] T. Caniard, et al., Phys. Rev. Lett. 99, 110801 (2007). [31] M. Cerdonio, et al., Phys. Rev. Lett. 87, 031101 (2001). [32] X. W. Xu, et al., Phys. Rev. A 93, 023827 (2016). [33] N. R. Bernier, et al., Nat. Commun. 8, 604 (2017) [34] V. Peano, et al., Phys. Rev. X 5, 031011 (2015). [35] G. A. Peterson, et al., Phys. Rev. X 7, 031001 (2017) [36] Control of interference phenomena in multimode
optome-chanical systems for the realisation of nonreciprocal de-vices through the control of the driving phases has been exploited also in Refs. [37–39], which are however based
0 0.25 0.5 0.75 1 5 . 0 0.75 1 0 0.25 (mm) (mm) (mm) (mm) (mm ) (mm ) (mm ) (mm ) (a) (c) (d) (b) 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 5 . 0 0.75 1 0 0.25 5 . 0 0.75 1 0 0.25 0 0.25 0.5 0.75 1
FIG. 11: Image (grey shape) of the electrodes used for realizing constructive interference, (a) and (b), and for de-structive interference, (c) and (d), overlapped with the mode shapes of the doublet obtained from finite element simula-tion. From these images we have calculated the effective ar-eas defined in Eq. (B8) and obtained G1 = 116.4 V m−1 and G2 = −109.6 V m−1 for the upper electrode configurations, and G1 = 117.0 V m−1 and G2 = 113.9 V m−1 for the lower electrode configurations. These values are in very good agree-ment with the values given in the main text and obtained as best-fit parameters of the measured output spectra.
on configurations different from the one demonstrated here.
[37] K. Fang, et al., Nat. Phys. 13, 465 (2017). [38] Y. Li, et al., arXiv:1705.08635 [quant-ph] (2017). [39] M. C. Kuzyk and H. Wang, arXiv1705.04722 (2017). [40] C. Biancofiore, et al., Phys. Rev. A 84, 033814 (2011). [41] E. Zeuthen, Ph.D. thesis, University of Copenhagen
